GRADE 6: Jack Hille Middle School
DOMAIN
CLUSTER
Ratio &
Proportional
Relationships
The Number
System
• Understand ratio concepts and use ratio reasoning to solve problems.
• Apply and extend previous understandings of multiplication and division
to divide fractions by fractions.
• Compute fluently with multi-digit numbers and find common factors and
multiples.
• Apply and extend previous understandings of numbers to the system of
rational numbers.
Expressions &
Equations
• Apply and extend previous understandings of arithmetic to algebraic
expressions.
• Reason about and solve one-variable equations and inequalities.
• Represent and analyze quantitative relationships between dependent and
independent variables.
Geometry
• Solve real-world and mathematical problems involving area, surface area,
and volume.
Statistics &
Probability
• Develop understanding of statistical variability.
• Summarize and describe distributions.
Mathematical Practices
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
DOMAIN
CLUSTER
Ratio & Proportional Analysis
Understand ratio concepts and use ratio reasoning to solve
problems
STANDARD
One: Understand the concept of a ratio and use ratio language to describe a
ratio relationship between two quantities.
For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1,
because for every 2 wings there was 1 beak.” “For every vote candidate A
received, candidate C received nearly three votes.”
Two: Understand the concept of a unit rate a/b associated with a ratio a:b with
b ≠ 0, and use rate language in the context of a ratio relationship.
For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so
there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers,
which is a rate of $5 per hamburger.”
Three: Use ratio and rate reasoning to solve real-world and mathematical
problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams,
double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number
measurements, find missing values in the tables, and plot the pairs of
values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and
constant speed. For example, if it took 7 hours to mow 4 lawns, then at
that rate, how many lawns could be mowed in 35 hours? At what rate
were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity
means 30/100 times the quantity); solve problems involving finding the
whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and
transform units appropriately when multiplying or dividing quantities.
Assessment Examples
Standard One Summative Assessment: Analyze the ratio of dollars to other forms of currency.
Formative Assessment:
 Interpret a currency table
 Compare and contrast values
 Express ratio in multiple representations
 Select appropriate algorithm
Standard Two Summative Assessment: Determine unit rate to analyze which is the best buy.
Support your answer with evidence.
Formative Assessment:
 Explain “unit rate”
 Select appropriate algorithm
 Calculate unit rate
 Support with numerical evidence and explanation
Standard Three Summative Assessment: An American news reporter has asked for your
assistance. Convert Olympic records from metric system to the customary system.
Formative Assessment:
 Distinguish between the metric and customary system
 Identify units for length, capacity, and mass
 Read a conversion table
 Interpret a conversion table
 Apply dimensional analysis
Instructional Strategies
Proportional reasoning is a process that requires instruction and practice. It does not develop over time
on its own. Grade 6 is the first of several years in which students develop this multiplicative thinking.
Examples with ratio and proportion must involve measurements, prices and geometric contexts, as
well as rates of miles per hour or portions per person within contexts that are relevant to sixth graders.
Experience with proportional and non-proportional relationships, comparing and predicting ratios, and
relating unit rates to previously learned unit fractions will facilitate the development of proportional
reasoning. Although algorithms provide efficient means for finding solutions, the cross- product
algorithm commonly used for solving proportions will not aid in the development of proportional
reasoning. Delaying the introduction of rules and algorithms will encourage thinking about
multiplicative situations instead of indiscriminately applying rules.
Students develop the understanding that ratio is a comparison of two numbers or quantities. Ratios
that are written as part-to-whole are comparing a specific part to the whole. Fractions and percents are
examples of part-to-whole ratios. Fractions are written as the part being identified compared to the
whole amount. A percent is the part identified compared to the whole (100). Provide students with
multiple examples of ratios, fractions and percents of this type. For example, the number of girls in the
class (12) to the number of students in the class (28) is the ratio 12 to 28.
Percents are often taught in relationship to learning fractions and decimals. This cluster indicates that
percents are to be taught as a special type of rate. Provide students with opportunities to find percents
in the same ways they would solve rates and proportions.
Part-to-part ratios are used to compare two parts. For example, the number of girls in the class (12)
compared to the number of boys in the class (16) is the ratio the ratio 12 to 16. This form of ratios is
often used to compare the event that can happen to the event that cannot happen.
Rates, a relationship between two units of measure, can be written as ratios, such as miles per hour,
ounces per gallon and students per bus. For example, 3 cans of pudding cost $2.48 at Store A and 6
cans of the same pudding costs $4.50 at Store B. Which store has the better buy on these cans of
pudding? Various strategies could be used to solve this problem:
 A student can determine the unit cost of 1 can of pudding at each store and compare.
 A student can determine the cost of 6 cans of pudding at Store A by doubling $2.48.
 A student can determine the cost of 3 cans of pudding at Store B by taking 1⁄2 of $4.50.
Using ratio tables develops the concept of proportion. By comparing equivalent ratios, the concept of
proportional thinking is developed and many problems can be easily solved.
Store A
3 cans
$2.48
6 cans
4.96
Store B
6 cans
$4.50
3 cans
$2.25
Students should also solve real-life problems involving measurement units that need to be converted.
Representing these measurement conversions with models such as ratio tables, t-charts or double
number line diagrams will help students internalize the size relationships between same system
measurements and relate the process of converting to the solution of a ratio.
Multiplicative reasoning is used when finding the missing element in a proportion. For example, use 2
cups of syrup to 5 cups of water to make fruit punch. If 6 cups of syrup are used to make punch, how
many cups of water are needed?
2/5 = 6/x
Recognize that the relationship between 2 and 6 is 3 times; 2 · 3 = 6
To find x, the relationship between 5 and x must also be 3 times. 3 · 5 = x, therefore,
x = 15
2/5= 6/15
The final proportion.
Other ways to illustrate ratios that will help students see the relationships follow. Begin written
representation of ratios with the words “out of” or “to” before using the symbolic notation of the colon
and then the fraction bar; for example, 3 out of 7, 3 to 5, 6:7 and then 4/5.
Use skip counting as a technique to determine if ratios are equal.
Labeling units helps students organize the quantities when writing proportions.
3 eggs/2 cups of flour = z eggs/8 cups of flour
Using hue/color intensity is a visual way to examine ratios of part-to-part. Students can compare the
intensity of the color green and relate that to the ratio of colors used. For example, have students mix
green paint into white paint in the following ratios: 1 part green to 5 parts white, 2 parts green to 3 parts
white, and 3 parts green to 7 parts white. Compare the green color intensity with their ratios.
Instructional Resources/Tools
100 grids (10 x 10) for modeling percents
Ratio tables – to use for proportional reasoning
Bar Models – for example, 4 red bars to 6 blue bars as a visual representation of a ratio and then
expand the number of bars to show other equivalent ratios
Common Misconceptions
Fractions and ratios may represent different comparisons. Fractions always express a part-to-whole
comparison, but ratios can express a part-to-whole comparison or a part-to-part comparison.
Even though ratios and fractions express a part-to-whole comparison, the addition of ratios and the
addition of fractions are distinctly different procedures. When adding ratios, the parts are added, the
wholes are added and then the total part is compared to the total whole. For example, (2 out of 3 parts)
+ (4 out of 5 parts) is equal to six parts out of 8 total parts (6 out of 8) if the parts are equal. When
dealing with fractions, the procedure for addition is based on a common
denominator: (2/3) + (4/5) = (10/15 + (12/15) which is equal to (22/15). Therefore, the addition process for
ratios and for fractions is distinctly different.
Often there is a misunderstanding that a percent is always a natural number less than or equal to 100.
Provide examples of percent amounts that are greater than 100%, and percent amounts that are less
1%.
DOMAIN
CLUSTER
The Number System
Apply and extend previous understandings of multiplication and
division to divide fractions by fractions.
STANDARD One: Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction models
and equations to represent the problem.
For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction
model to show the quotient; use the relationship between multiplication and
division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general,
(a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people
share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a
cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and
area 1/2 square mi?
Assessment Examples
Standard One Summative Assessment: You know the portion size per person and the weight of
the roast, determine the number of people you can feed. Justify your answer.
Formative Assessment:
 Represent decimals as fractions
 Multiply fractions
 Simplify fractions
 Apply division algorithm to fractions
 Use calculated amount to determine appropriate answer
 Justify answer
Instructional Strategies
Computation with fractions is best understood when it builds upon the familiar understandings of
whole numbers and is paired with visual representations. Solve a simpler problem with whole
numbers, and then use the same steps to solve a fraction divided by a fraction. Looking at the
problem through the lens of “How many groups?” or “How many in each group?” helps visualize
what is being sought.
For example: 12÷3 means; How many groups of three would make 12? Or how many in each of 3
groups would make 12? Thus 7/2 ÷ 1/ 4 can be solved the same way. How many groups of ¼
make 7/2 ?
Or, how many objects in a group when 7/2 fills one fourth?
Creating the picture that represents this problem makes seeing and proving the solutions easier.
Add pictorial representation
Set the problem in context and represent the problem with a concrete or pictorial model. 5/4
÷ ½.
5/4 cups of nuts filled ½ of a container. How many cups of nuts will fill the entire container.
How many cups of nuts will fill the entire container?
Teaching “invert and multiply” without developing an understanding of why it works first leads to
confusion as to when to apply the shortcut.
Learning how to compute fraction division problems is one part, being able to relate the problems
to real-world situations is important. Providing opportunities to create stories for fraction problems
or writing equations for situations is needed.
Instructional Resources/Tools
Models for Multiplying and Dividing Fractions
http://www.learner.org/courses/learningmath/number/session9/part_a/area_division.html
National Library of Virtual Manipulatives:
Search: Fractions-Rectangle Multiplication
Use this virtual manipulative to graphically demonstrate, explore, and practice multiplying fractions.
Common Misconceptions
Students may believe that dividing by 1/2 is the same as dividing in half. Dividing by half means to find
how many 1/2s there are in a quantity, whereas, dividing in half means to take a quantity and split it
into two equal parts. Thus 7 divided by 1/2 = 14 and 7 divided in half equals 3 1/2 .
DOMAIN
CLUSTER
The Number System
Compute fluently with multi-digit numbers and find common
factors and multiples.
STANDARD Two: Fluently divide multi-digit numbers using the standard algorithm.
Three: Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
Four: Find the greatest common factor of two whole numbers less than or
equal to 100 and the least common multiple of two whole numbers less than or
equal to 12. Use the distributive property to express a sum of two whole
numbers 1–100 with a common factor as a multiple of a sum of two whole
numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Assessment Examples
Standard Two Summative Assessment: Fluently Divide multi-digit numbers
Formative Assessment:
 Distinguish between divisor, dividend, and quotient
 Apply algorithm
 Multiplication fluency
 Subtraction fluency
 Using zeros for place holders
Standard Three Summative Assessment: Fluently add, subtract, multiply, and divide multi-digit
decimals
Formative Assessment:
 Place value (addition and subtraction)
 Renaming to add and subtract
 Apply algorithm
 Multiplication fluency
 Division fluency
 Place value (multiplication and division)
Standard Four Summative Assessment: Apply the distributive process in reverse to determine the
greatest common factor
Formative Assessment:
 Find the greatest common factor of two whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less than or equal to 12.
 Recognizes that finding the greatest common factor is the distributive process in reverse
Instructional Strategies
As students study whole numbers in the elementary grades, a foundation is laid in the conceptual
understanding of each operation. Discovering and applying multiple strategies for computing creates
connections, which evolve into the proficient use of standard algorithms.
Fluency with an algorithm denotes an ability that is efficient, accurate, appropriate and flexible. Division
was introduced in Grade 3 conceptually, as the inverse of multiplication. In Grade 4, division continues
using place-value strategies, properties of operations, the relationship with multiplication, area models,
and rectangular arrays to solve problems with one digit divisors. In Grade 6, fluency with the
algorithms for division and all operations with decimals is developed.
Fluency is something that develops over time; practice should be given over the course of the year as
students solve problems related to other mathematical studies. Opportunities to determine when to use
paper pencil algorithms, mental math or a computing tool is also a necessary skill and should be
provided in problem solving situations.
Greatest common factor and least common multiple are usually taught as a means of combining
fractions with unlike denominators. This cluster builds upon the previous learning of the multiplicative
structure of whole numbers, as well as prime and composite numbers in Grade 4. Although the
process is the same, the point is to become aware of the relationships between numbers and their
multiples. For example, consider answering the question: “If two numbers are multiples of four, will the
sum of the two numbers also be a multiple of four?” Being able to see and write the relationships
between numbers will be beneficial as further algebraic understandings are developed. Another focus
is to be able to see how the GCF is useful in expressing the numbers using the distributive property,
(36 + 24) = 12(3+2), where 12 is the GCF of 36 and 24. This concept will be extended in Expressions
and Equations as work progresses from understanding the number system and solving equations to
simplifying and solving algebraic equations in Grade 7
Instructional Resources/Tools
Exploring GCF
http://www.math.com/school/subject1/lessons/S1U3L2GL.html
Common Misconceptions
DOMAIN
CLUSTER
The Number System
Apply and extend previous understandings of numbers to the
systems of rational numbers.
STANDARD Five: Understand that positive and negative numbers are used together to
describe quantities having opposite directions or values (e.g., temperature
above/below zero, elevation above/below sea level, credits/debits,
positive/negative electric charge); use positive and negative numbers to
represent quantities in real-world contexts, explaining the meaning of 0 in each
situation.
Six: Understand a rational number as a point on the number line. Extend
number line diagrams and coordinate axes familiar from previous grades to
represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite
sides of 0 on the number line; recognize that the opposite of the opposite of a
number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in
quadrants of the coordinate plane; recognize that when two ordered pairs differ
only by signs, the locations of the points are related by reflections across one or
both axes.
c. Find and position integers and other rational numbers on a horizontal or
vertical number line diagram; find and position pairs of integers and other
rational numbers on a coordinate plane.
Seven: Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of
two numbers on a number line diagram. For example, interpret –3 > –7 as a
statement that –3 is located to the right of –7 on a number line oriented from left
to right.
b. Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write –3 degrees C > –7 degrees C to express the
fact that –3 degrees C is warmer than –7 degrees C.
c. Understand the absolute value of a rational number as its distance from 0 on
the number line; interpret absolute value as magnitude for a positive or negative
quantity in a real-world situation. For example, for an account balance of –30
dollars, write –30 = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For
example, recognize that an account balance less than –30 dollars represents a
debt greater than 30 dollars.
Eight: Solve real-world and mathematical problems by graphing points in all
four quadrants of the coordinate plane. Include use of coordinates and absolute
value to find distances between points with the same first coordinate or the
same second coordinate.
Assessment Examples
Standard Five Summative Assessment: Use positive and negative numbers to represent
quantities in real-world contexts, explaining the meaning of 0 in each situation.
Formative:
 Debit/Credit
 Negative/Positive
 Above/Below
 Deposit/Withdrawl
 Over/Under (Text Minutes)
Standard Six Summative Assessment: Analyze the coordinates on a graph to draw conclusions.
(Think of a real life example that is represented by the plotted data) 1. Scuba Diving –
Relationship depth to pressure 2. Flying - Increase altitude, decrease air pressure)
Formative:
 Recognize the directionality associated with negative and positive coordinates
 Identify quadrants
 Represent points on the line and in the plane with negative number coordinates.
 Explain that the opposite of an opposite is itself
Standard Seven Summative Assessment: Manage a check register and support your final
balance with evidence using the words “debit” and “credit”.
Formative:
 Define “debit” and “credit”
 Translate statements and identify “debit” and “credit”
 Use appropriate operation fluently
 Support final balance with evidence
Standard Eight Summative Assessment: Given two sets of coordinates with the same first or
second coordinate, representing two towns, find the distance.
Formative:
 Find coordinates on a graph
 Use absolute value
 Complete subtraction algorithm to find distance
Instructional Strategies
The purpose of this cluster is to begin study of the existence of negative numbers, their relationship to
positive numbers, and the meaning and uses of absolute value. Starting with examples of
having/owing and above/below zero sets the stage for understanding that there is a mathematical way
to describe opposites.
Students should already be familiar with the counting numbers (positive whole numbers and zero), as
well as with fractions and decimals (also positive). They are now ready to understand that all numbers
have an opposite. These special numbers can be shown on vertical or horizontal number lines, which
then can be used to solve simple problems. Demonstration of understanding of positives and
negatives involves translating among words, numbers and models: given the words “7 degrees below
zero,” showing it on a thermometer and writing -7; given -4 on a number line, writing a real-life example
and mathematically -4. Number lines also give the opportunity to model absolute value as the distance
from zero.
Simple comparisons can be made and order determined. Order can also be established and written
mathematically: -3° C > -5° C or -5° C < -3° C. Finally, absolute values should be used to relate
contextual problems to their meanings and solutions.
Using number lines to model negative numbers, prove the distance between opposites, and
understand the meaning of absolute value easily transfers to the creation and usage of four-quadrant
coordinate grids. Points can now be plotted in all four quadrants of a coordinate grid. Differences
between numbers can be found by counting the distance between numbers on the grid. Actual
computation with negatives and positives is handled in Grade 7.
Instructional Resources/Tools
Common Misconceptions
DOMAIN
Expressions & Equations
Apply and extend previous understandings of arithmetic to algebraic equations.
CLUSTER
STANDARD One: Write and evaluate numerical expressions involving whole-number exponents.
Two: Write, read, and evaluate expressions in which letters(variables) stand for
numbers.
a. Write expressions that record operations with numbers and with letters(variables)
standing for numbers.
For example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product,
factor, quotient, coefficient); view one or more parts of an expression as a single
entity.
For example, describe the expression 2 (8 + 7) as a product of two factors; view (8
+ 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions
that arise from formulas used in real-world problems. Perform arithmetic operations,
including those involving whole-number exponents, in the conventional order when
there are no parentheses to specify a particular order (Order of Operations).
For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface
area of a cube with sides of length s = 1/2.
Three: Apply the properties of operations to generate equivalent expressions.
For example, apply the distributive property to the expression 3 (2 + x) to produce
the equivalent expression 6 + 3x; apply the distributive property to the expression
24x + 18y to produce the equivalent expression 6 (4x +3y); apply properties of
operations to y + y + y to produce the equivalent expression 3y.
Four: Identify when two expressions are equivalent (i.e., when the two expressions
name the same number regardless of which value is substituted into them). For
example, the expressions y + y + y and 3y are equivalent because they name the
same number regardless of which number y stands
Assessment Examples
Standard One Summative Assessment: Translate verbal phrases into mathematical expressions.
Formative:
 Interpret words associated with operations
 Match word with visual representation
 Apply properties
 Place numbers in correct place in relation to the operations symbol
Standard Two Summative Assessment: Given a scenario of purchasing a suitcase, and the given
values determine which is the best purchase to maximize capacity.
Formative:
 Perform arithmetic operations including exponents
 Apply formula and analyze calculations
 Determine which suitcase has maximum capacity
Standard Three Summative Assessment: Given a movie ticket scenario where there is one given,
one unknown and a constant calculate total cost for a family.
Formative:
 Translate
 Set up expression with appropriate operations and grouping symbols
 Apply the distributive property
Standard Four Summative Assessment: To commemorate September 11th, Soldier Field
displayed a flag that covered the entire field. Express the perimeter of the flag in two different yet
equivalent expressions.
Formative:
 Recognize the attributes of a rectangle
 Distinguish between area and perimeter
 Apply distributive property to determine two expressions
Instructional Strategies
The skills of reading, writing and evaluating expressions are essential for future work with expressions
and equations, and are a Critical Area of Focus for Grade 6. In earlier grades, students added
grouping symbols ( ) to reduce ambiguity when solving equations. Now the focus is on using ( ) to
denote terms in an expression or equation. Students should now focus on what terms are to be solved
first rather than invoking the PEMDAS rule. Likewise, the division symbol (3 ÷5) was used and should
now be replaced with a fraction bar (3/5). Less confusion will occur as students write algebraic
expressions and equations if x represents only variables and not multiplication. The use of a dot (.) or
parentheses between number terms is preferred.
Provide opportunities for students to write expressions for numerical and real-world situations. Write
multiple statements that represent a given algebraic expression. For example, the expression x – 10
could be written as “ten less than a number,” “a number minus ten,” “the temperature fell ten degrees,”
“I scored ten fewer points than my brother,” etc. Students should also read an algebraic expression
and write a statement.
Through modeling, encourage students to use proper mathematical vocabulary when discussing
terms, factors, coefficients, etc. Provide opportunities for students to write equivalent expressions, both
numerically and with variables. For example, given the expression x + x + x + x + 4•2, students could
write 2x + 2x + 8 or some other equivalent expression. Make the connection to the simplest form of this
expression as 4x + 8. Because this is a foundational year for building the bridge between the concrete
concepts of arithmetic and the abstract thinking of algebra, using hands-on materials (such as algebra
tiles, counters, unifix cubes, "Hands on Algebra") to help students translate between concrete
numerical representations and abstract symbolic representations is critical.
Provide expressions and formulas to students, along with values for the variables so students can
evaluate the expression. Evaluate expressions using the order of operations with and without
parentheses. Include whole-number exponents, fractions, decimals, etc. Provide a model that shows
step-by-step thinking when simplifying an expression. This demonstrates how two lines of work
maintain equivalent algebraic expressions and establishes the need to have a way to review and justify
thinking.
Provide a variety of expressions and problem situations for students to practice and deepen their skills.
Start with simple expressions to evaluate and move to more complex expressions. Likewise start with
simple whole numbers and move to fractions and decimal numbers. The use of negatives and
positives should mirror the level of introduction in Grade 6 The Number System; students are
developing the concept and not generalizing operation rules.
The use of technology can assist in the exploration of the meaning of expressions. Many calculators
will allow you to store a value for a variable and then use the variable in expressions. This enables the
student to discover how the calculator deals with expressions like x2, 5x, xy, and 2(x + 5).
Instructional Resources/Tools
Virtual Algebra Tiles
http://illuminations.nctm.org/ActivityDetail.aspx?ID=216
Algebra Game: Late Delivery
http://www.bbc.co.uk/education/mathsfile/shockwave/games/postie.html
Common Misconceptions
Many of the misconceptions when dealing with expressions stem from the misunderstanding/reading
of the expression. For example, knowing the operations that are being referenced with notation like, x3,
4x, 3(x + 2y) is critical. The fact that x3 means x␣x␣x, means x times x times x, not 3x or 3 times x; 4x
means 4 times x or x+x+x+x, not forty-something. When evaluating 4x when x = 7, substitution does
not result in the expression meaning 47. Use of the “x” notation as both the variable and the operation
of multiplication can complicate this understanding.
DOMAIN
Expressions & Equations
CLUSTER
Reason about and solve one-variable equations
STANDARD Five: Understand solving an equation or inequality as a process of answering a
question: which values from a specified set, if any, make the equation or inequality
true? Use substitution to determine whether a given number in a specified set
makes an equation or inequality true.
Six: Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an
unknown number, or, depending on the purpose at hand, any number in a specified
set.
Seven: Solve real-world and mathematical problems by writing and solving
equations of the form x + p = q and px = q for cases in which p, q and x are all
nonnegative rational numbers.
Eight: Write an inequality of the form x > c or x < c to represent a constraint or
condition in a real-world or mathematical problem. Recognize that inequalities of the
form x > c or x < c have infinitely many solutions; represent solutions of such
inequalities on number line diagrams.
Assessment Examples
Standard Five Summative Assessment: Determine if an inequality or equation is “true” before
solving. Defend or justify your thinking.
Formative:
 Distinguish between an equation and an inequality
 Know rational number relationships
 Know absolute value
Standard Six Summative Assessment: There are 36 inches in a yard, write an expression to find
the number of yards in x inches.
Formative:
 Understand that fraction bar means division
 Distinguish between divisor and dividend in fraction form
Standard Seven Summative Assessment: If 240 sixth and seventh graders attend a dance and
130 of them are sixth graders, how many are seventh graders?
Formative:
 Define variables
 Translate from a verbal sentence to a mathematical equation
 Solve applying appropriate operations
Standard Eight Summative Assessment: A movie is rated PG 13, therefore a parent must
accompany anyone under 13. Write an inequality to represent your ability to get into the show
versus your older sibling’s ability to get into the show without a parent. Represent the inequalities
on a number line.
Formative:
 Appropriate use of inequality signs
 Translate from verbal phrase to inequality
 Show a graphic representation on a number line representing greater than or equal to
versus greater than
Instructional Strategies
The skill of solving an equation must be developed conceptually before it is developed procedurally.
This means that students should be thinking about what numbers could possibly be a solution to the
equation before solving the equation. For example, in the equation x + 21 = 32 students know that 21
+ 9 = 30 therefore the solution must be 2 more than 9 or 11, so x = 11.
Provide multiple situations in which students must determine if a single value is required as a solution,
or if the situation allows for multiple solutions. This creates the need for both types of equations (single
solution for the situation) and inequalities (multiple solutions for the situation).
Solutions to equations should not require using the rules for operations with negative numbers since
the conceptual understanding of negatives and positives is being introduced in Grade 6. When working
with inequalities, provide situations in which the solution is not limited to the set of positive whole
numbers but includes rational numbers. This is a good way to practice fractional numbers and
introduce negative numbers. Students need to be aware that numbers less than zero could be part of a
solution set for a situation. As an extension to this concept, certain situations may require a solution
between two numbers. For example, a problem situation may have a solution that requires more than
10 but not greater than 25. Therefore, the exploration with students as to what this would look like both
on a number line and symbolically is a reasonable extension.
The process of translating between mathematical phrases and symbolic notation will also assist
students in the writing of equations/inequalities for a situation. This process should go both ways;
Students should be able to write a mathematical phrase for an equation. Additionally, the writing of
equations from a situation or story does not come naturally for many students. A strategy for assisting
with this is to give students an equation and ask them to come up with the situation/story that the
equation could be referencing.
Instructional Resources/Tools
Common Misconceptions
DOMAIN
CLUSTER
Expressions & Equations
Represent and analyze quantitative relationships between
dependent and independent variables.
STANDARD
Nine: Use variables to represent two quantities in a real-world problem that change
in relationship to one another; write an equation to express one quantity, thought of
as the dependent variable, in terms of the other quantity, thought of as the
independent variable. Analyze the relationship between the dependent and
independent variables using graphs and tables, and relate these to the equation.
For example, in a problem involving motion at constant speed, list and graph
ordered pairs of distances and times, and write the equation d = 65t to represent the
relationship between distance and time.
Assessment Examples
Standard Nine Summative Assessment: If a race car travels at 200 mph, how far can it travel d
distance in t hours. Support using an appropriately labeled coordinate graph.
Formative:
 Identify factors and constant
 Set up distance equals rate x time formula
 Identify independent and dependent variable
 Create a tabular representation
 Construct graph
 Plot points to show relationship of distance to time
Instructional Strategies
The goal is to help students connect the pieces together, this can be done by having students use
multiple representations for the mathematical relationship. Students need to be able to translate freely
among the story, words (mathematical phrases), models, tables, graphs and equations. They also
need to be able to start with any of the representations and develop the others.
Provide multiple situations for the student to analyze and determine what unknown is dependent on
the other components. For example, how far I travel is dependent on the time and rate that I am
traveling.
Throughout the expressions and equations domain in Grade 6, students need to have an
understanding of how the expressions or equations relate to situations presented, as well as the
process of solving them.
The use of technology, including computer apps, CBLs, and other hand-held technology allows the
collection of real- time data or the use of actual data to create tables and charts. It is valuable for
students to realize that although real- world data often is not linear, a line sometimes can model the
data .
Instructional Resources/Tools
NCTM Illuminations: Pedal Power
http://illuminations.nctm.org/LessonDetail.aspx?id=L586
Common Misconceptions
Students may misunderstand what the graph represents in context. For example, that moving up or
down on a graph does not necessarily mean that a person is moving up or down.
DOMAIN
CLUSTER
Geometry
Solve real-world problems and mathematical problems involving
area, surface area, and volume.
STANDARD
One: Find the area of right triangles, other triangles, special quadrilaterals, and
polygons by composing into rectangles or decomposing into triangles and other
shapes; apply these techniques in the context of solving real-world and
mathematical problems.
Two: Find the volume of a right rectangular prism with fractional edge lengths by
packing it with unit cubes of the appropriate unit fraction edge lengths, and show
that the volume is the same as would be found by multiplying the edge lengths of
the prism. Apply the formulas V = l w h and V = b h to find volumes
of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
Three: Draw polygons in the coordinate plane given coordinates for the vertices;
use coordinates to find the length of a side joining points with the same first
coordinate or the same second coordinate. Apply these techniques in the context of
solving real-world and mathematical problems.
Four: Represent three-dimensional figures using nets made up of rectangles and
triangles, and use the nets to find the surface area of these figures. Apply these
techniques in the context of solving real-world and mathematical problems.
Assessment Examples
Standard One Summative Assessment: Given the area of an entire classroom, use a floor plan to
determine the area of the unused space.
Formative:
 Know formulas for determining area of triangles and quadrilaterals
 Apply formulas to determine occupied space
 Subtract occupied space from total area
 Informally prove your statement
Standard Two Summative Assessment: Calculate the number of cases of food that can be stored
in the corner of Sam’s Club given a limited volume and the case size with fractional edge lengths.
Formative:
 Formula for volume
 Multiply fractions
 Calculate volume of space
 Calculate volume of case
 Divide volume of space by case
 Apply logical reasoning
Standard Three Summative Assessment: You want to paint the mascot on the gymnasium doors.
The problem is you only have a small drawing. Given the coordinates (1,2) and (3,2) of your scale
drawing of the mascot calculate the enlarged image using a scale factor of 3 to find the
coordinates of the larger figure and use that data to find the length of the mascot.
Formative:
 Locate the given coordinates on the plane
 Apply scale factor
 Locate new coordinates
 Use data to find length of mascot
Standard Four Summative Assessment: Given the dimensions of a skateboard launch ramp find
the amount of paint needed given that one quart covers 100 square feet.
Formative:
 Draw a net and label all dimensions
 Fold the net to be certain it accurately represents the launch ramp
 Open and calculate the area of each polygon
 Total the areas
 Solve using division algorithm, proportional reasoning or algebraic equation
Instructional Strategies
It is very important for students to continue to physically manipulate materials and make connections to
the symbolic and more abstract aspects of geometry. Exploring possible nets should be done by taking
apart (unfolding) three-dimensional objects. This process is also foundational for the study of surface
area of prisms. Building upon the understanding that a net is the two-dimensional representation of the
object, students can apply the concept of area to find surface area. The surface area of a prism is the
sum of the areas for each face.
Multiple strategies can be used to aid in the skill of determining the area of simple two-dimensional
composite shapes. A beginning strategy should be to use rectangles and triangles, building upon
shapes for which they can already determine area to create composite shapes. This process will
reinforce the concept that composite shapes are created by joining together other shapes, and that the
total area of the two-dimensional composite shape is the sum of the areas of all the parts.
add pictorial representation
A follow-up strategy is to place a composite shape on grid or dot paper. This aids in the decomposition
of a shape into its foundational parts. Once the composite shape is decomposed, the area of each part
can be determined and the sum of the area of each part is the total area.
add pictorial representation
Fill prisms with cubes of different edge lengths (including fractional lengths) to explore the relationship
between the length of the repeated measure and the number of units needed. An essential
understanding to this strategy is the volume of a rectangular prism does not change when the units
used to measure the volume changes. Since focus in Grade 6 is to use fractional lengths to measure,
if the same object is measured using one centimeter cubes and then measured using half centimeter
cubes, the volume will appear to be eight times greater with the smaller unit. However, students need
to understand that the value or the number of cubes is greater but the volume is the same.
Instructional Resources/Tools
Online dot paper
http://illuminations.nctm.org/lessons/DotPaper.pdf#search=%22dot paper%22
Lessons on area
http://illuminations.nctm.org/LessonDetail.aspx?ID=L580
Common Misconceptions
Students may believe that the orientation of a figure changes the figure. In Grade 6, some students still
struggle with recognizing common figures in different orientations. For example, a square rotated 45°
is no longer seen as a square and instead is called a diamond. This impacts students’ ability to
decompose composite figures and to appropriately apply formulas for area. Providing multiple
orientations of objects within classroom examples and work is essential for students to overcome this
misconception.
DOMAIN
Statistics & Probability
CLUSTER
Develop understanding of statistical variability.
STANDARD One: Recognize a statistical question as one that anticipates variability in the data
related to the question and accounts for it in the answers.
For example, “How old am I?” is not a statistical question, but “How old are the
students in my school?” is a statistical question because one anticipates variability
in students’ ages.
Two: Understand that a set of data collected to answer a statistical question has a
distribution, which can be described by its center, spread, and overall shape.
Three: Recognize that a measure of center for a numerical data set summarizes all
of its values with a single number, while a measure of variation describes how its
values vary with a single number.
Assessment Examples
Standard One Summative Assessment: Determine which would be a statistical question and
support your choice with evidence and explanation.
Formative:
 Distinguish statistical and non-statistical questions
 Write a statistical question and defend its validity
Standard Two Summative Assessment: Analyze a set of class scores from a recent assessment.
Find mean, median and mode. Determine range and inter-quartile range and deviation. Display
the data. Use the data display to describe the story of the data.
Formative:
 Distinguish between mean, median and mode
 Determine mean, median, and mode
 Distinguish, range, inter-quartile range and deviation
 Display data
 Determine range, inter-quartile range and deviation
 Analyze and describe the meaning of the data
Standard Three Summative Assessment: Given statistical scenarios determine when it is
appropriate to report mean or range using measures of central tendency or spread.
 Analyze data
 Defend choice
Instructional Strategies
Grade 6 is the introduction to the formal study of statistics for students. Students need multiple
opportunities to look at data to determine and word statistical questions. Data should be analyzed from
many sources, such as organized lists, box-plots, bar graphs and stem-and-leaf plots. This will help
students begin to understand that responses to a statistical question will vary, and that this variability is
described in terms of spread and overall shape.
At the same time, students should begin to relate their informal knowledge of mean, mode and median
to understand that data can also be described by single numbers. The single value for each of the
measures of center (mean, median or mode) and measures of spread (range, inter-quartile range,
mean absolute deviation) is used to summarize the data. Given measures of center for a set of data,
students should use the value to describe the data in words. The important purpose of the number is
not the value itself, but the interpretation it provides for the variation of the data. Interpreting different
measures of center for the same data develops the understanding of how each measure sheds a
different light on the data. The use of a similarity and difference matrix to compare mean, median,
mode and range may facilitate understanding the distinctions of purpose between and among the
measures of center and spread.
Include activities that require students to match graphs and explanations, or measures of center and
explanations prior to interpreting graphs based upon the computation measures of center or spread.
The determination of the measures of center and the process for developing graphical representation
is the focus of the cluster “Summarize and describe distributions” in the Statistics and Probability
domain for Grade 6. Classroom instruction should integrate the two clusters.
Instructional Resources/Tools
Newspaper and magazine graphs for analysis of the spread, shape and variation of data
Illuminations: Numerical & Categorical Data
http://illuminations.nctm.org/LessonDetail.aspx?id=U116
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/category_g_3_t_5.html
Common Misconceptions
Students may believe all graphical displays are symmetrical. Exposing students to graphs of various
shapes will show this to be false.
The value of a measure of center describes the data, rather than a value used to interpret and
describe the data.
DOMAIN
Statistics & Probability
Summarize and describe distributions
CLUSTER
STANDARD Four: Display numerical data in plots on a number line, including dot plots,
histograms, and box plots.
Five: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was
measured and its
units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability
(inter-quartile range
and/or mean absolute deviation), as well as describing any overall pattern and any
striking
deviations from the overall pattern with reference to the context in which the data
were gathered.
d. Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
Assessment Examples
Standard Four Summative Assessment: Use data from standards one, two, and three to display
numerical data in plots on a number line, including dot plots, histograms, and box plots.
Formative:
 Distinguish between plots on a number line, including dot plots, histograms, and box plots
 Given data sets choose appropriate display
Standard Five Summative Assessment: See Assessments for Standards One to Three
Instructional Strategies
This cluster builds on the understandings developed in the Grade 6 cluster “Develop understanding of
statistical variability.” Students have analyzed data displayed in various ways to see how data can be
described in terms of variability. Additionally, in Grades 3-5 students have created scaled picture and
bar graphs, as well as line plots. Now students learn to organize data in appropriate representations
such as box plots (box-and-whisker plots), dot plots, and stem-and-leaf plots. Students need to display
the same data using different representations. By comparing the different graphs of the same data,
students develop understanding of the benefits of each type of representation.
Further interpretation of the variability comes from the range and center-of-measure numbers. Prior to
learning the computation procedures for finding mean and median, students will benefit from concrete
experiences.
To find the median visually and kinesthetically, students should reorder the data in ascending or
descending order, then place a finger on each end of the data and continue to move toward the center
by the same increments until the fingers touch. This number is the median.
The concept of mean (concept of fair shares) can be demonstrated visually and kinesthetically by
using stacks of linking cubes. The blocks are redistributed among the towers so that all towers have
the same number of blocks. Students should not only determine the range and centers of measure, but
also use these numbers to describe the variation of the data collected from the statistical question
asked. The data should be described in terms of its shape, center, spread (range) and inter-quartile
range or mean absolute deviation (the absolute value of each data point from the mean of the data
set). Providing activities that require students to sketch a representation based upon given measures
of center and spread and a context will help create connections between the measures and real-life
situations.
Continue to have students connect contextual situations to data to describe the data set in words prior
to computation. Therefore, determining the measures of spread and measures of center
mathematically need to follow the development of the conceptual understanding. Students should
experience data which reveals both different and identical values for each of the measures. Students
need opportunities to explore how changing a part of the data may change the measures of center and
measure of spread. Also, by discussing their findings, students will solidify understanding of the
meanings of the measures of center and measures of variability, what each of the measures do and do
not tell about a set of data, all leading to a better understanding of their usage.
Using graphing calculators to explore box plots (box-and-whisker plots) removes the time intensity
from their creation and permits more time to be spent on the meaning. It is important to use the interquartile range in box plots when describing the variation of the data. The mean absolute deviation
describes the distance each point is from the mean of that data set. Patterns in the graphical displays
should be observed, as should any outliers in the data set. Students should identify the attributes of
the data and know the appropriate use of the attributes when describing the data. Pairing contextual
situations with data and its box-and-whisker plot is essential.
Instructional Resources/Tools
Graphing calculators may also be used for creating lists and displaying the data. Guidelines for
Assessment and Instruction in Statistics Education (GAISE) report. American Statistical Association
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/category_g_3_t_5.html
Common Misconceptions
Students often use words to help them recall how to determine the measures of center. However,
student’s lack of understanding of what the measures of center actually represent tends to confuse
them. Median is the number in the middle, but that middle number can only be determined after the
data entries are arranged in ascending or descending order. Mode is remembered as the “most,” and
often students think this means the largest value, not the “most frequent” entry in the set. Vocabulary is
important in mathematics, but conceptual understanding is equally as important. Usually the mean,
mode, or median have different values, but sometimes those values are the same.